Retract Rational Fields

Abstract

Let $k$ be an infinite field. The notion of retract $k$-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1. Let $k\subset K\subset L$ be fields. If $K$ is retract $k$-rational and $L$ is retract $K$-rational, then $L$ is retract $k$-rational. Theorem 2. For any finite group $G$ containing an abelian normal subgroup $H$ such that $G/H$ is a cyclic group, for any complex representation $G \to GL(V)$, the fixed field $\bm{C}(V)^G$ is retract $\bm{C}$-rational. Theorem 3. If $G$ is a finite group, then all the Sylow subgroups of $G$ are cyclic if and only if $\bm{C}_{\alpha}(M)^G$ is retract $\bm{C}$-rational for all $G$-lattices $M$, for all short exact sequences $\alpha : 0 \to \bm{C}^{\times} \to M_{\alpha} \to M \to 0$. Because the unramified Brauer group of a retract $\bm{C}$-rational field is trivial, Theorem 2 and Theorem 3 generalize previous results of Bogomolov and Barge respectively (see Theorem \ref{t5.9} and Theorem \ref{t6.1}).